$ A = \left[\begin{array}{rr}1 & 1 \\ 3 & 5\end{array}\right]$ $ C = \left[\begin{array}{rr}4 & 5 \\ 1 & 5\end{array}\right]$ What is $ A C$ ?
Answer: Because $ A$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A C = \left[\begin{array}{rr}{1} & {1} \\ {3} & {5}\end{array}\right] \left[\begin{array}{rr}{4} & \color{#DF0030}{5} \\ {1} & \color{#DF0030}{5}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{4}+{1}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{4}+{1}\cdot{1} & ? \\ {3}\cdot{4}+{5}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{4}+{1}\cdot{1} & {1}\cdot\color{#DF0030}{5}+{1}\cdot\color{#DF0030}{5} \\ {3}\cdot{4}+{5}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{4}+{1}\cdot{1} & {1}\cdot\color{#DF0030}{5}+{1}\cdot\color{#DF0030}{5} \\ {3}\cdot{4}+{5}\cdot{1} & {3}\cdot\color{#DF0030}{5}+{5}\cdot\color{#DF0030}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}5 & 10 \\ 17 & 40\end{array}\right] $